Matrix-free construction of HSS representations using adaptive randomized sampling Xiaoye Sherry Li [email protected]Gustavo Chavez, Pieter Ghysels, Chris Gorman, Yang Liu Lawrence Berkeley National Laboratory Chris Gorman, UC Santa Barbara Randomized Numerical Linear Algebra and Applications X.S. Li Faster Linear Solvers Sept. 27 1 / 28
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Matrix-free construction of HSS representations usingadaptive randomized sampling
Gustavo Chavez, Pieter Ghysels, Chris Gorman, Yang LiuLawrence Berkeley National Laboratory
Chris Gorman, UC Santa Barbara
Randomized Numerical Linear Algebra and Applications
X.S. Li Faster Linear Solvers Sept. 27 1 / 28
Acknowledgement
This research was supported by the Exascale Computing Project(http://www.exascaleproject.org), a joint project of the U.S. Departmentof Energys Office of Science and National Nuclear Security Administration,responsible for delivering a capable exascale ecosystem, including software,applications, and hardware technology, to support the nations exascalecomputing imperative.Project Number: 17-SC-20-SC
X.S. Li Faster Linear Solvers Sept. 27 2 / 28
Hierarchical matrix approximation
• Same mathematical foundation as FMM (Greengard-Rokhlin’87), putin matrix form:• Diagonal block (“near field”) represented exactly• Off-diagonal block (“far field”) approximated via low-rank format
FMM Algebraicseparability of Green’s function low rank off-diagonal
• Sampling is handy, but still needs more mathematical insight to makeit robust and efficient.
• Preconditioner appears to be robust [IPDPS 2017]• Works well for problems where AMG has slow convergence, e.g.,
indefinite problems.• More parallelizable than ILU, fewer parameters to tune.
• More research• Dynamic load balancing.• Communication analysis for sparse solvers.• Rank analysis of different application problems.• Good ordering and hierarchical clustering / partitioning to reduce
off-diagonal rank.• Not all problems compress well in HSS, look into other formats.
X.S. Li Faster Linear Solvers Sept. 27 23 / 28
THANK YOU !
X.S. Li Faster Linear Solvers Sept. 27 24 / 28
HSS matrix – ULV factorization
ULV-like factored form (U and V ∗ unitary, L triangular)
Γ1;b↔2;t
I
Ω1I
Ω2
[Γ3;b↔4;tΓ5;b↔6;t
]Ω3
Ω4Ω5
Ω6
AQ∗3
Q∗4Q∗5
Q∗6
[
ΓT3;b↔4;t
ΓT5;b↔6;t
]IQ∗1
IQ∗2
ΓT1;b↔2;t
=
L3
0 L4
(Ω1L4,3)t (Ω1L3,4)t L1
L5
0 0 L6
(Ω2L6,5)t (Ω2L5,6)t L2
(Ω1L4,3)b (Ω1L3,4)b W1;bQ∗1;t B1,2V
∗2
[V∗5 Q∗5;t V∗5 Q∗5;b
V∗6 Q∗6;t V∗6 Q∗6;b
] [IQ∗2
]D0
B2,1V∗1
[V∗3 Q∗3;t V∗3 Q∗3;b
V∗4 Q∗4;t V∗4 Q∗4;b
] [IQ∗1
](Ω2L6,5)b (Ω2L5,6)b W2;bQ
∗2;t
X.S. Li Faster Linear Solvers Sept. 27 25 / 28
Themes
Many research areas for exascale computing: https://exascaleproject.org
• Algorithms with lower arithmetic & communication complexity.Multilevel algorithms:• Multigrid• Fast Multipole Method (FMM)• Hierarchical matrices – algebraic generalization of FMM,
applicable to broader classes of problems
• Parallel algorithms and codes for machines with million-wayparallelism, hierarchical organization.• Distributed memory• Manycore nodes: 100s of lightweight cores, accelerators,
Many research areas for exascale computing: https://exascaleproject.org
• Algorithms with lower arithmetic & communication complexity.Multilevel algorithms:• Multigrid• Fast Multipole Method (FMM)• Hierarchical matrices – algebraic generalization of FMM,
applicable to broader classes of problems
• Parallel algorithms and codes for machines with million-wayparallelism, hierarchical organization.• Distributed memory• Manycore nodes: 100s of lightweight cores, accelerators,
• indefinite, ill-conditioned, nonsymmetric (e.g. those from multiphysics,multiscale simulations)
• Where can be used?• Stand-alone solver.• Good for multiple right-hand sides.• Precondition Krylov solvers.• Coarse-grid solver in multigrid. (e.g., Hypre)• In nonlinear solver. (e.g., SUNDIALS)• Solving interior eigenvalue prolems.• ...